Straight-line backbone curve

In this work we propose a class of problems in nonlinear vibrations related to avoiding undesirable hysteresis and jump phenomena and offer sample conservative systems for which the backbone curve is a straight vertical line.

► The backbone curve is a feature of an unforced nonlinear conservative oscillator.
► The backbone curve describes its amplitude–frequency relationship.
► We present a variety of nonlinear oscillators with a straight vertical backbone curve.
► When forced, such oscillators do not exhibit unwanted hysteresis and jump phenomena.
► We derive an exact closed-form O.D.E. yielding a straight-line backbone curve.